Licensed under the Creative Commons attribution-noncommercial license, http://creativecommons.org/licenses/by-nc/3.0/. Please share and remix noncommercially, mentioning its origin.
CC-BY_NC

Produced in R version 3.2.1 using pomp version 0.71.1.

Objectives

  1. display a published case study using plug-and-play methods with non-trivial model complexities
  2. demonstrate the use of profile likelihood in scientific inference
  3. discuss the interpretation of parameter estimates
  4. emphasize the need to allow for extra sources of stochasticity in modeling

Measles revisited

Motivation: challenges in inference from disease dynamics

  • Understanding, forecasting, managing epidemiological systems increasingly depends on models
  • Dynamic models can be used to test causal hypotheses
  • Real epidemiological systems:
    • are nonlinear
    • are stochastic
    • are nonstationary
    • evolve in continuous time
    • have hidden variables
    • can be measured only with (large) error
  • Dynamics of infectious disease outbreaks illustrate this well

Outline

  • revisit classic measles data set
  • ask questions about:
    • measles extinction and recolonization
    • transmission rates
    • seasonality
    • resupply of susceptibles
  • use a model that
    1. expresses our current understanding of measles dynamics
    2. cannot be fit by existing likelihood-based methods
  • examine data from large and small towns using the same model
  • does our perspective on this disease change?
  • what bigger lessons can we learn regarding inference for dynamical systems?

He, Ionides, & King, J. R. Soc. Interface (2010)

Data sets

  • Twenty towns
    • population sizes: 2k–3.4M
    • Weekly case reports, 1950–1963
    • Annual birth records and population sizes, 1944–1963

Continuous-time Markov process model

  • \(B(t) = \text{birth rate, from data}\)
  • \(N(t) = \text{population size, from data}\)

  • overdispersed binomial measurement model: \(\mathrm{cases}_t\,\vert\,I{\to}R=z_t \sim {\mathrm{Normal}\left(\rho\,z_t,\rho\,(1-\rho)\,z_t+(\psi\,\rho\,z_t)^2\right)}\)

  • entry into susceptible class: \[\mu_{BS}(t) = (1-c)\,B(t-\tau)+c\,\delta(t-t_0)\,\int_{t-1}^{t}\,B(t-\tau-s)\,ds\]

  • force of infection \[\mu_{SE}(t) = \tfrac{\beta(t)}{N(t)}\,(I+\iota)\,\zeta(t)\]

  • \(c = \text{cohort effect}\)
  • \(\tau = \text{school-entry delay}\)
  • \(\beta(t) = \text{school-term transmission}\)
  • \(\iota = \text{imported infections}\)

  • \(\zeta(t) = \text{white noise with intensity}\,\sigma_{SE}\)

Fitting procedure

  • a large Latin-hypercube design was used to initiate searches
  • iterated filtering to maximize the likelihood
  • point estimates of all parameters for 20 cities
  • profile likelihoods to quantify uncertainty in London and Hastings

Imported infections

\[\mu_{SE}=\frac{\beta(t)}{N(t)}\,(I+\iota)\,\zeta(t)\]

Seasonality

Notable findings

Cohort effect

\(R_0\)

Birth delay

Reporting rate

Predicted vs observed critical community size

Parameter estimates

pop R0 amplitude LP IP alpha iota rho psi sigmaSE
Halesworth 2170 33.100 0.381 7.870 2.280 0.948 0.00912 0.754 0.641 0.0748
Lees 4250 29.700 0.153 8.510 2.050 0.968 0.03110 0.612 0.681 0.0802
Mold 6410 21.400 0.271 5.930 1.780 1.040 0.01450 0.131 2.870 0.0544
Dalton.in.Furness 10600 28.300 0.203 5.480 1.980 0.989 0.03860 0.455 0.818 0.0779
Oswestry 11000 52.900 0.339 10.300 2.720 1.040 0.02980 0.631 0.476 0.0699
Northwich 18300 30.100 0.423 8.510 3.010 0.948 0.06020 0.795 0.402 0.0857
Bedwellty 28900 24.700 0.160 6.820 3.030 0.937 0.03960 0.311 0.951 0.0611
Consett 39100 35.900 0.200 9.070 2.660 1.010 0.07310 0.650 0.406 0.0712
Hastings 65700 34.200 0.299 7.000 5.440 1.000 0.18600 0.695 0.396 0.0955
Cardiff 245000 34.400 0.223 9.860 3.090 0.996 0.14100 0.602 0.270 0.0539
Bradford 294000 32.100 0.236 8.510 3.360 0.991 0.24400 0.599 0.190 0.0451
Hull 302000 38.900 0.221 9.180 5.460 0.968 0.14200 0.582 0.256 0.0636
Nottingham 307000 22.600 0.157 5.720 3.690 0.982 0.17000 0.609 0.258 0.0380
Bristol 443000 26.800 0.203 6.190 4.940 1.010 0.44100 0.626 0.201 0.0392
Leeds 510000 47.800 0.267 9.480 10.900 1.000 1.25000 0.666 0.167 0.0778
Sheffield 515000 33.100 0.313 7.230 6.380 1.020 0.85300 0.649 0.175 0.0428
Manchester 704000 32.900 0.290 11.100 6.940 0.965 0.59000 0.550 0.161 0.0551
Liverpool 802000 48.100 0.305 7.900 9.800 0.978 0.26300 0.494 0.136 0.0533
Birmingham 1120000 43.400 0.428 8.520 11.600 1.010 0.34300 0.544 0.178 0.0611
London 3390000 56.800 0.554 13.100 12.500 0.976 2.90000 0.488 0.116 0.0878
r NA 0.444 0.293 0.349 0.946 0.108 0.93200 -0.203 -0.931 -0.3170

\(r = \mathrm{cor}(\log{\hat\theta},\log{N_{1950}})\)

Extrademographic stochasticity

\[\mu_{SE}=\frac{\beta(t)}{N(t)}\,(I+\iota)\,\zeta(t)\]

Practicum

Preliminaries

We’ll load the packages we’ll need, and set the random seed, to allow reproducibility.

set.seed(594709947L)
require(ggplot2)
theme_set(theme_bw())
require(plyr)
require(reshape2)
require(magrittr)
require(pomp)
stopifnot(packageVersion("pomp")>="0.70-1")

Data and covariates

Now we’ll load the data and covariates. The data are measles reports from 20 cities in England and Wales. We also have information on the population sizes and birth-rates in these cities; we’ll treat these variables as covariates.

baseurl <- "http://kinglab.eeb.lsa.umich.edu/SBIED/"
download.file(paste0(baseurl,"data/twentycities.rda"),destfile="./twentycities.rda")
load("twentycities.rda")
measles %>% 
  mutate(year=as.integer(format(date,"%Y"))) %>%
  subset(town=="London" & year>=1950 & year<1964) %>%
  mutate(time=(julian(date,origin=as.Date("1950-01-01")))/365.25+1950) %>%
  subset(time>1950 & time<1964, select=c(time,cases)) -> dat
demog %>% subset(town=="London",select=-town) -> demog

Let’s plot the data and covariates.

dat %>% ggplot(aes(x=time,y=cases))+geom_line()

demog %>% melt(id="year") %>%
  ggplot(aes(x=year,y=value))+geom_point()+
  facet_wrap(~variable,ncol=1,scales="free_y")

demog %>% 
  summarize(
    time=seq(from=min(year),to=max(year),by=1/12),
    pop=predict(smooth.spline(x=year,y=pop),x=time)$y,
    birthrate=predict(smooth.spline(x=year+0.5,y=births),x=time-4)$y
    ) -> covar
plot(pop~time,data=covar,type='l')
points(pop~year,data=demog)

plot(birthrate~time,data=covar,type='l')
points(births~year,data=demog)

plot(birthrate~I(time-4),data=covar,type='l')
points(births~I(year+0.5),data=demog)

The partially observed Markov process model

The (unobserved) process model

Let’s evaluate the hypothesis that these data were generated by an SEIR model. The SEIR model is a compartmental model that, diagrammatically, looks as follows.

\(b = \text{births}\)
\(S = \text{susceptibles}\)
\(E = \text{exposed, incubating}\)
\(I = \text{infectious}\)
\(R = \text{recovered}\)

We require a simulator for this model. The following implements a simulator.

rproc <- Csnippet("
  double beta, br, seas, foi, dw, births;
  double rate[6], trans[6];
  
  // cohort effect
  if (fabs(t-floor(t)-251.0/365.0) < 0.5*dt) 
    br = cohort*birthrate/dt + (1-cohort)*birthrate;
  else 
    br = (1.0-cohort)*birthrate;

  // term-time seasonality
  t = (t-floor(t))*365.25;
  if ((t>=7&&t<=100) || (t>=115&&t<=199) || (t>=252&&t<=300) || (t>=308&&t<=356))
      seas = 1.0+amplitude*0.2411/0.7589;
    else
      seas = 1.0-amplitude;

  // transmission rate
  beta = R0*(gamma+mu)*seas;
  // expected force of infection
  foi = beta*pow(I+iota,alpha)/pop;
  // white noise (extrademographic stochasticity)
  dw = rgammawn(sigmaSE,dt);

  rate[0] = foi*dw/dt;  // stochastic force of infection
  rate[1] = mu;             // natural S death
  rate[2] = sigma;        // rate of ending of latent stage
  rate[3] = mu;             // natural E death
  rate[4] = gamma;        // recovery
  rate[5] = mu;             // natural I death

  // Poisson births
  births = rpois(br*dt);
  
  // transitions between classes
  reulermultinom(2,S,&rate[0],dt,&trans[0]);
  reulermultinom(2,E,&rate[2],dt,&trans[2]);
  reulermultinom(2,I,&rate[4],dt,&trans[4]);

  S += births   - trans[0] - trans[1];
  E += trans[0] - trans[2] - trans[3];
  I += trans[2] - trans[4] - trans[5];
  R = pop - S - E - I;
  W += (dw - dt)/sigmaSE;  // standardized i.i.d. white noise
  C += trans[4];           // true incidence
")

In the above, \(C\) represents the true incidence, i.e., the number of new infections occurring over an interval. Since recognized measles infections are quarantined, we argue that most infection occurs before case recognition so that true incidence is a measure of the number of individuals progressing from the I to the R compartment in a given interval.

We complete the process model definition by specifying the distribution of initial unobserved states. The following codes assume that the fraction of the population in each of the four compartments is known.

initlz <- Csnippet("
  double m = pop/(S_0+E_0+I_0+R_0);
  S = nearbyint(m*S_0);
  E = nearbyint(m*E_0);
  I = nearbyint(m*I_0);
  R = nearbyint(m*R_0);
  W = 0;
  C = 0;
")

The measurement model

We’ll model both under-reporting and measurement error. We want \(\mathbb{E}[\text{cases}|C] = \rho\,C\), where \(C\) is the true incidence and \(0<\rho<1\) is the reporting efficiency. We’ll also assume that \(\mathrm{Var}[\text{cases}|C] = \rho\,(1-\rho)\,C + (\psi\,\rho\,C)^2\), where \(\psi\) quantifies overdispersion. Note that when \(\psi=0\), the variance-mean relation is that of the binomial distribution. To be specific, we’ll choose \(\text{cases|C} \sim f(\cdot|\rho,\psi,C)\), where \[f(c|\rho,\psi,C) = \Phi(c+\tfrac{1}{2},\rho\,C,\rho\,(1-\rho)\,C+(\psi\,\rho\,C)^2)-\Phi(c-\tfrac{1}{2},\rho\,C,\rho\,(1-\rho)\,C+(\psi\,\rho\,C)^2),\] where \(\Phi(x,\mu,\sigma^2)\) is the c.d.f. of the normal distribution with mean \(\mu\) and variance \(\sigma^2\).

The following computes \(\mathbb{P}[\text{cases}|C]\).

dmeas <- Csnippet("
  double m = rho*C;
  double v = m*(1.0-rho+psi*psi*m);
  double tol = 1.0e-18;
  if (cases > 0.0) {
      lik = pnorm(cases+0.5,m,sqrt(v)+tol,1,0)-pnorm(cases-0.5,m,sqrt(v)+tol,1,0)+tol;
  } else {
    lik = pnorm(cases+0.5,m,sqrt(v)+tol,1,0)+tol;
  }
")

The following codes simulate \(\text{cases} | C\).

rmeas <- Csnippet("
  double m = rho*C;
  double v = m*(1.0-rho+psi*psi*m);
  double tol = 1.0e-18;
  cases = rnorm(m,sqrt(v)+tol);
  if (cases > 0.0) {
    cases = nearbyint(cases);
  } else {
    cases = 0.0;
  }
")

Constructing the pomp object

We put all the model components together with the data in a call to pomp:

dat %>% 
  pomp(t0=with(dat,2*time[1]-time[2]),
       time="time",
       rprocess=euler.sim(rproc,delta.t=1/365.25),
       initializer=initlz,
       dmeasure=dmeas,
       rmeasure=rmeas,
       covar=covar,
       tcovar="time",
       zeronames=c("C","W"),
       statenames=c("S","E","I","R","C","W"),
       paramnames=c("R0","mu","sigma","gamma","alpha","iota",
                    "rho","sigmaSE","psi","cohort","amplitude",
                    "S_0","E_0","I_0","R_0")
       ) -> m1

The following codes plot the data and covariates together.

m1 %>% as.data.frame() %>% 
  melt(id="time") %>%
  ggplot(aes(x=time,y=value))+
  geom_line()+
  facet_grid(variable~.,scales="free_y")

He et al. (2010) estimated the parameters of this model. The full set is included in the R code accompanying this document, where they are read into a data frame called mles.

mles %>% subset(town=="London") -> mle
paramnames <- c("R0","mu","sigma","gamma","alpha","iota",
                "rho","sigmaSE","psi","cohort","amplitude",
                "S_0","E_0","I_0","R_0")
mle %>% extract(paramnames) %>% unlist() -> theta
mle %>% subset(select=-c(S_0,E_0,I_0,R_0)) %>%
  knitr::kable(row.names=FALSE)
town loglik loglik.sd mu delay sigma gamma rho R0 amplitude alpha iota cohort psi sigmaSE
London -3804.9 0.16 0.02 4 28.9 30.4 0.488 56.8 0.554 0.976 2.9 0.557 0.116 0.0878

Verify that we get the same likelihood as He et al. (2010).

require(foreach)
require(doMC)

registerDoMC()

set.seed(998468235L,kind="L'Ecuyer")
mcopts <- list(preschedule=FALSE,set.seed=TRUE)

foreach(i=1:4,
        .packages="pomp",
        .options.multicore=mcopts) %dopar% {
  pfilter(m1,Np=10000,params=theta)
} -> pfs
logmeanexp(sapply(pfs,logLik),se=TRUE)
##                          se 
## -3801.0666251     0.9129177

Simulations at the MLE.

m1 %>% 
  simulate(params=theta,nsim=9,as.data.frame=TRUE,include.data=TRUE) %>%
  ggplot(aes(x=time,y=cases,group=sim,color=(sim=="data")))+
  guides(color=FALSE)+
  geom_line()+facet_wrap(~sim,ncol=2)

m1 %>% 
  simulate(params=theta,nsim=100,as.data.frame=TRUE,include.data=TRUE) %>%
  subset(select=c(time,sim,cases)) %>%
  mutate(data=sim=="data") %>%
  ddply(~time+data,summarize,
        p=c(0.05,0.5,0.95),q=quantile(cases,prob=p,names=FALSE)) %>%
  mutate(p=mapvalues(p,from=c(0.05,0.5,0.95),to=c("lo","med","hi")),
         data=mapvalues(data,from=c(TRUE,FALSE),to=c("data","simulation"))) %>%
  dcast(time+data~p,value.var='q') %>%
  ggplot(aes(x=time,y=med,color=data,fill=data,ymin=lo,ymax=hi))+
  geom_ribbon(alpha=0.2)

Parameter transformations

The parameters are constrained to be positive, and some of them are constrained to lie between \(0\) and \(1\). We can turn the likelihood maximization problem into an unconstrained maximization problem by transforming the parameters. The following Csnippets implement such a transformation and its inverse.

toEst <- Csnippet("
  Tmu = log(mu);
  Tsigma = log(sigma);
  Tgamma = log(gamma);
  Talpha = log(alpha);
  Tiota = log(iota);
  Trho = logit(rho);
  Tcohort = logit(cohort);
  Tamplitude = logit(amplitude);
  TsigmaSE = log(sigmaSE);
  Tpsi = log(psi);
  TR0 = log(R0);
  to_log_barycentric (&TS_0, &S_0, 4);
")

fromEst <- Csnippet("
  Tmu = exp(mu);
  Tsigma = exp(sigma);
  Tgamma = exp(gamma);
  Talpha = exp(alpha);
  Tiota = exp(iota);
  Trho = expit(rho);
  Tcohort = expit(cohort);
  Tamplitude = expit(amplitude);
  TsigmaSE = exp(sigmaSE);
  Tpsi = exp(psi);
  TR0 = exp(R0);
  from_log_barycentric (&TS_0, &S_0, 4);
")


pomp(m1,toEstimationScale=toEst,
     fromEstimationScale=fromEst,
     statenames=c("S","E","I","R","C","W"),
     paramnames=c("R0","mu","sigma","gamma","alpha","iota",
                  "rho","sigmaSE","psi","cohort","amplitude",
                  "S_0","E_0","I_0","R_0")) -> m1

Exercises

Reformulate the model

Modify the He et al. (2010) model to remove the cohort effect. Run simulations and compute likelihoods to convince yourself that the resulting codes agree with the original ones for cohort = 0.

Now modify the transmission seasonality to use a sinusoidal form. How many parameters must you use? Fixing the other parameters at their MLE values, compute and visualize a profile likelihood over these parameters.

Extrademographic stochasticity

Set the extrademographic stochasticity parameter \(\sigma_{SE}=0\), set \(\alpha=1\), and fix \(\rho\) and \(\iota\) at their MLE values, then maximize the likelihood over the remaining parameters. How do your results compare with those at the MLE? Compare likelihoods but also use simulations to diagnose differences between the models.

References

He, D., E. L. Ionides, and A. A. King. 2010. Plug-and-play inference for disease dynamics: Measles in large and small populations as a case study. Journal of The Royal Society Interface 7:271–283.